Differentiate twice, i.e find and solve to find any inflection points.
Here's what I got for the 1st and 2nd derivative:
f'(x) = [be^(-1/2 b^2 (a-x)^2)] [a-x] / sqrt[2*pi]
f"(x) = [be^(-1/2 b^2 (a-x)^2)] [a^2 b^2 + x^2 b^2 -2axb^2 - 1] / sqrt [2 *pi]
Too many variables. I don't know if derivatives are correct or how to solve for 0.
I would introduce new symbols:
The differentiations are less cumbersome and it's easier to see how to factorise and solve the equation . Substitute as required the old symbols once you've solved for x.
Note that learning latex will make your posts easier to read. This will probably increase the pool of people willing to reply, which will lead to getting faster replies.